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Quadrilaterals Notes: Class 9 Maths Chapter 8

Class 9 Maths Chapter 8 Quadrilaterals covers the angle sum property of a quadrilateral, properties of a parallelogram, conditions for a quadrilateral to be a parallelogram, and the Mid-point Theorem.

Overview of Topics & Subtopics (Click to Jump)

1. Angle Sum Property & Types of Quadrilaterals

Short Note:

• The sum of the angles of a quadrilateral is 360°.

• Special quadrilaterals include the parallelogram, rectangle, rhombus, square and trapezium, each with distinctive properties of sides, angles and diagonals.

2. Properties of a Parallelogram

Short Note:

• A diagonal of a parallelogram divides it into two congruent triangles.

• In a parallelogram: opposite sides are equal, opposite angles are equal, and the diagonals bisect each other.

3. Conditions for a Quadrilateral to be a Parallelogram

Short Note:

• A quadrilateral is a parallelogram if: opposite sides are equal, OR opposite angles are equal, OR diagonals bisect each other, OR a pair of opposite sides is equal and parallel.

4. The Mid-point Theorem

Short Note:

• The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.

• Converse: The line drawn through the mid-point of one side of a triangle, parallel to another side, bisects the third side.

NCERT Textbook Exercise Questions

The angles of a quadrilateral are in the ratio 3:5:9:13. Find all the angles.
Answer: Let angles = 3x, 5x, 9x, 13x. Sum = 360°: 30x = 360° ⇒ x = 12°. Angles are 36°, 60°, 108°, 156°.
Show that the diagonals of a square are equal and bisect each other at right angles.
Answer: Using congruence of triangles formed by the diagonals of square ABCD (SAS/SSS rules with all sides equal and all angles 90°), it can be shown AC = BD, and using the parallelogram property the diagonals bisect each other; further congruence shows they intersect at 90°.
ABCD is a rhombus. Show that diagonal AC bisects ∠A as well as ∠C, and diagonal BD bisects ∠B as well as ∠D.
Answer: Since all sides of a rhombus are equal, ΔABC ≅ ΔADC (SSS), so ∠BAC = ∠DAC and ∠BCA = ∠DCA, i.e. AC bisects ∠A and ∠C. Similarly BD bisects ∠B and ∠D.
ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that D is the mid-point of AC.
Answer: Since M is the mid-point of AB and MD || BC, by the converse of the Mid-point Theorem, D must be the mid-point of AC.
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Class 9 › Mathematics › Quadrilaterals
Chapter notesRead the chapterPractice MCQsCheck what you rememberChapter test10 questions on one pageOnline testTest 1, 2 and 3, one question at a timePYQPrevious year questionsStudy materialPDF notes
Topics in this chapter: Angle Sum Property & Types of QuadrilateralsProperties of a ParallelogramConditions for a Quadrilateral to be a ParallelogramThe Mid-point Theorem